1. Ale with Mixed Elements 10 – 14 September 2007 Ale with Mixed Elements Ale with Mixed Elements C. Aymard, J. Flament, J.P. Perlat

2. Ale with Mixed Elements 10 – 14 September 2007 Contents Introduction Description of Lagrangian step  Treatment of the multi-material cells  Elastic-Plastic treatment Description of advection step  Criteria and mesh smoothing algorithms  Interface reconstruction in multi-material cells  Material mesh reconstruction  Remapping Some results Conclusion

3. Ale with Mixed Elements 10 – 14 September 2007 Introduction Because hydrocodes with lagrangian scheme lack of robustness specially on thin meshes, an ALE formulation with multi-material elements has been implemented to perform simulations with small sizes cells : We introduce ALE blocks with several materials in the same mesh (mixed and pure cells) which can interact with each other or with pure Lagrangian by slide lines. In a same simulation, we can track accurately material boundaries (sliding, void opening and closure) and high material deformation. = Ale block with M1/M2/M3 materials = Lagrangian

4. Ale with Mixed Elements 10 – 14 September 2007 Notations In Practice : Splitting method Governing equations Main features 1 – Mesh smoothing under criteria 2 – Remapping : nodal or cell values ( ,  u,  e) from the old mesh onto the new mesh Time derivative : Lagrangian : Ale Reference system : Advection phase : Lagrangian phase Lagrangian phase : u = v u(x, t) : material velocity v(x, t) : mesh velocity

5. Ale with Mixed Elements 10 – 14 September 2007 Splitting method Lagrangian step : with multi-material cells  Use the classical Wilkins second order scheme  Special treatment in multi-material cells : ð Assumption of equal material volumetric strain for all materials ð Interface reconstruction Advection step :  Mesh smoothing under criteria : a new grid is defined.  Remapping phase : the material quantities are interpolated from the old configuration onto the new one.  Material by material, after material mesh reconstruction ð Dukowicz method (intersection old mesh / new mesh), second order accuracy

6. Ale with Mixed Elements 10 – 14 September 2007 Lagrangian step with multi-material cells Treatment of the multi-material cell : Assumption : Only one velocity for all material :  imply the relation closure for volume fraction in a cell :  equal material volumetric strain rate for all materials  And so with the internal energy balance equation, we have the average pressure definition Note Volume fraction ● More robust multi-material cell treatment : Relaxation pressure we perform an iterative method after Lagrangian step (without volume variation) to balance material pressure in a cell. equal volumetric strain for all materials

7. Ale with Mixed Elements 10 – 14 September 2007 Lagrangian step with multi-material cells Ale t = 0.2 s Sod shock tube problem ALE Lagrange material pressure profiles P = 0.59 P1 = 0.593 P2 = 0.4 Mat1 Mat2 t = 0. s Material P = 1. P = 0.1

8. Ale with Mixed Elements 10 – 14 September 2007 Elastic-plastic treatment For each elastic material  characterized by mechanical characteristics (  , Y  ), we compute the deviatoric stress tensor: On the Ale Block : Same deformation Assumption For the momentum conservative equation Deviatoric stress tensor Stress tensor Lagrangian step with multi-material cells Average quantities The average stress is the average of the individual material stresses weighted by the volume fraction.

9. Ale with Mixed Elements 10 – 14 September 2007 Advection step : Mesh smoothing under criteria Object : improve the mesh quality at the end of the lagrangian step, by finding some ideal position for the nodes, ideal in the sense of minimizing element distortions. We use zones definitions which allow user to specify and to adjust locally the mesh smoothing. We define nodal remapping criteria to identify the nodes which need to be relaxed :  For boundary nodes : distance criterion  For internal nodes : volume criterion, angular criterion … The boundary nodes are moved by specific algorithms (specially for the sliding material boundaries) ;  To apply an equidistant node distribution along the boundary,  To preserve the shape of the boundary. The new position of internal nodes is calculated from classical iterative mesh smoothing algorithms, with a possibility of remapping constraint :  Winslow’s equipotential relaxation method  Simple average method

10. Ale with Mixed Elements 10 – 14 September 2007 Multi-material cell advection step Remapping phase : Interface Material Reconstruction in mixed cells  Multi-material cells are characterized by their volume fractions.  To track the material interface, we use the Young’s method : the slope is given by the gradient of the volume fractions field :  Materials are ordered in a cell by locating each material mass center along the normal direction.  Only one normal for all interfaces : onion skin method Material mesh reconstruction  From the old grid, we build nodal-cell connectivity for each material : ð We create the nodes between the material interface and the mesh edge. ð The common nodes are merged. ð The new material lagrangian mesh is build with non structured connectivity mat2 mat3 New common nodes mat2 mat1

11. Ale with Mixed Elements 10 – 14 September 2007 Multi-material cell advection step Remapping step : we perform the second order Dukowicz method (intersection old mesh / new mesh) : Interfaces Reconstruction For each material, we  generate an old material mesh,  map material cell quantities from this old material mesh onto the new grid. Nodal quantities velocities are mapped from the old dual grid (global mesh) onto the new dual one. Material Mesh generation Interface reconstruction in mixed cell mat1 mat2 mat3 mat2 Continuity of the material mesh mat2 accurate gradient 2 nd order accuracy New grid Mapping by intersection old mesh / new mesh Material quantities : p .. Nodal quantities : v… New common nodes

12. Ale with Mixed Elements 10 – 14 September 2007 Some Results Instability with 3 materials : Taylor bar impact in both Lagrangian and ALE formulation Calculation of a dynamic friction experiment

13. Ale with Mixed Elements 10 – 14 September 2007 Instability with 3 Materials   P    P    P  100 25 Eulerian formulation ALE formulation Lagrangian formulation

14. Ale with Mixed Elements 10 – 14 September 2007 Instability with 3 materials Ale Lagrangian Lagrangian Interface very distorted. The calculation stops.

15. Ale with Mixed Elements 10 – 14 September 2007 Instability with 3 materials Ale Eulerian Same robustness of eulerian approach

16. Ale with Mixed Elements 10 – 14 September 2007 ALE Taylor bar impact Lagrange ALE P1 Y P1 time Good deformation Loss of accuracy and reducing the time step. Elements severely distorted (unphysical mesh distortion and hourglassing) Plastic strain 

17. Ale with Mixed Elements 10 – 14 September 2007 Dynamic friction experiment ALE = High Explosive / CONF Sliding with friction Sliding with opening Explosive Aluminium Sliding Steel Ale and Lagrangian mesh motion in the same framework

18. Ale with Mixed Elements 10 – 14 September 2007 Dynamic friction experiment Pure Lagrangian Lagrangian and ALE Comparison at 30  s  Good agreement with the Lagrangian deformation

19. Ale with Mixed Elements 10 – 14 September 2007 Dynamic friction experiment Lagrangian materials ALE block Zoom  ALE reduces mesh distortion in the mixed cells but the Ale boundary remains sharp.

20. Ale with Mixed Elements 10 – 14 September 2007 Conclusion Ale method with mixed cells : Lagrangian step with treatment of the multi-material cells ( equal volumetric strain assumption + relaxation pressure) Full 2D advection step with interface reconstruction in multi-material cells 2 different treatments for materials interfaces : Lagrangian interfaces with slidelines Multi materials cells.  Provides the accuracy of Lagrangian formulation and the robustness of the Eulerian Multi materials cells Future works : Extension to the 3D